model reduction of a logistic network based on ranking. The rank of ... [4], which has been a core component of Google Internet search engine in its early days, is ...
An appr oach to model r eduction of logistic networ ks based on r anking Ber nd Scholz-Reiter 1, Fabian Wir th2, Ser gey Dashkovskiy3, Michael Kosmykov3, Thomas Makuschewitz1, Michael Schönlein2 1
Planning and Control of Production Systems (PSPS), BIBA – Bremer Institut für Produktion und Logistik GmbH at the University of Bremen, Bremen, Germany 2 Institute of Mathematics, University of Würzburg, Würzburg, Germany 3 Centre of Industrial Mathematics, University of Bremen, Bremen, Germany
Abstr act Simulations or mathematical analysis of a real-world logistic network require a model. In this context two challenges occur for modelling: First, the model should represent the real-world logistic network in an accurate way. Second, it should foster simulations or analytical analysis to be conducted in a reasonable time. A large size is often a drawback of many models. In the case of logistic networks this drawback can be overcome by reducing the number of locations and transportation links of the graph model. In this paper we present an approach to model reduction of a logistic network based on ranking. The rank of a given location states the importance of the location for the whole network. In order to calculate the importance of a location we introduce an adaptation of the PageRank algorithm for logistic networks. The information about the rank and the structural relations between the locations are used for our approach to model reduction. Depending on the structural relation between locations we suggest three different approaches to obtain a model of lower size.
1
Introduction
Performance and competitiveness of a large-scale logistic network depend on the capability of the network to meet the expectations of the customers [1]. This capability is strongly connected to an effective management of the material flow within the network. The material flow is subject to the complex and often global structure of the network as well as to the dynamics of production and transportation processes [2]. In order to support the management, a better understanding of the dynamics related to the material flow and their consequences for the performance of the logistic network is required. In the literature several methods exist to analyse the material flow.
2
Three different methods can be utilized for the investigation of the material flow of a logistic network [3]. First, the material flow of the real-world network can be measured. Second, simulations can be carried out in order to analyse changes in the structure and dynamics of the logistic network for different scenarios. Third, mathematical methods can be applied in order to obtain a more precise understanding of the involved processes. Both, simulation and mathematical methods require usually a model of the real-world logistic network. Model development faces two major challenges. First, the model should exhibit almost the same properties compared to the real-world logistic network and second, it should be tailored to the applied methods in order to enable consolidated findings. The size of a model is often crucial for a successful application of a certain analysis method. A model of lower size facilitates simulations or the application of mathematical methods. Since, logistic networks often consist of a large number of locations and transportation connections between them a representative model of lower size is desired. Our approach to model reduction is based on a ranking scheme of the locations that takes the material flows within the network and the structure of the connections between the locations into account. For this purpose the PageRank algorithm [4], which has been a core component of Google Internet search engine in its early days, is used. The original ranking algorithm is extended by the results of a material flow analysis. A material flow analysis provides valuable information about the importance of the connections between the locations by analysing the quantities of material flow between the locations [5]. These quantities can be incorporated into the ranking algorithm in order to enhance the ranking. The adapted ranking algorithm provides in terms of its application to a logistic network a reasonable ranking [6], [7]. In order to derive a model of lower size we propose to focus on locations with a low importance for the network. According to their connections to other locations of the network we investigate three different approaches for model reduction. These approaches involve the exclusion and aggregation of individual locations as well as the exclusion of subparts of the network. The paper shows that by applying these approaches for model reduction representative models can be derived. The outline of the paper is as follows. In section 2 our proposed adaptation of the PageRank algorithm to logistic networks is presented. Three different approaches to model reduction based on the structural properties of the locations are introduced in section 3 and illustrated by examples. Section 4 summarise the findings of this paper and provides an outlook to future research.
2
Adaptation of the PageRank algorithm to logistic networks
Before we introduce a notion of importance rank of a logistic location in a network we describe the network itself as a model.
3
We model a logistic network as a directed graph in the following way. Let the logistic locations be numbered by 1,…,n and each of them be a node of this graph. There is an edge from node i to node j if there is a material, information or monetary flow from the ith to jth location. In our approach only the aggregated quantity of a material flow between the locations over a certain period of time is considered. Let a ij be a number quantifying this flow. In particular, if there is no flow from location i to location j then a ij=0. The matrix A=( a ij)i,j=1,…,n describes the interconnection structure and the flows of a given network and is called weighted adjacency matrix of the graph. For example the network in Figure 1 has the following weighted adjacency matrix: 0 0 0 7 0 0 4 5 0 A= 0 0 2 0 11 0 6 0 0
0 0 0 0 0 0 0 0 0 0
0 8 0 0
6 0 7 0
Fig.1. Weighted directed graph of a logistic network. The numbers in the circles are the numbers of nodes and the numbers near the edges are their weights.
The following types of matrices will be used in this paper: A matrix A is called column-normalized if for all i=1,…,n n
∑a j =1
ij
1, if there exists j such that aij ≠ 0 = 0, if for all j aij = 0 n
A matrix A is column-stochastic if
∑a
ij
= 1 for all i=1,…,n.
j =1
It is called primitive if there exists a positive integer k such that the matrix Ak has only positive elements. We call an adjacency matrix A irreducible if the corresponding graph is strongly connected, i.e., for every nodes i and j of the graph there exists a sequence of directed edges connecting i to j. Note that any primitive matrix is irreducible. Now we are ready to introduce the notion of importance of logistic locations in a network. In the sequel we call it rank of a node or location. We say that the rank of a certain location depends on the network structure, its position within the network and flows in this network. As the weighted adjacency matrix of a network contains the information about its structure and flows between its nodes we use it to define the importance of the nodes. To define the rank of logistic locations we use the idea of the PageRank [5], which was originally designed for the ranking web pages in the Internet. This idea and the algorithm for its calculation can be adapted in the following way to logistic networks. We say that the rank of a loca-
4
tion i depends on the flows a ij, from this node to other locations j=1,…,n as well as ranks of these locations The more important locations receive material from a given location the more important it is. In comparison with the original PageRank locations do not share their rank equally between their suppliers but rather proportionally to the flows from their suppliers described by a ij . I.e., the proportions a~ij aij
are a~ij =
. Such proportions were proposed in [9] as a modification of
∑ akj
k ∈I dj
PageRank. Thus the rank NRi of the node i should be calculated by n
NRi =
∑ a~ NR , ij
i = 1,..., n .
j
(1)
j =1
However we introduce a parameter α, 0
